• Communications of the Korean Mathematical Society
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• v.28 no.3
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• pp.425-432
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• 2013

Let S be a minimal surface of general type with $p_g(S)=q(S)=0$ having an involution ${\sigma}$ over the field of complex numbers. It is well known that if the bicanonical map ${\varphi}$ of S is composed with ${\sigma}$, then the minimal resolution W of the quotient $S/{\sigma}$ is rational or birational to an Enriques surface. In this paper we prove that the surface W of S with $K^2_S=5,6,7,8$ having an involution ${\sigma}$ with which the bicanonical map ${\varphi}$ of S is composed is rational. This result applies in part to surfaces S with $K^2_S=5$ for which ${\varphi}$ has degree 4 and is composed with an involution ${\sigma}$. Also we list the examples available in the literature for the given $K^2_S$ and the degree of ${\varphi}$.

• Journal of the Chungcheong Mathematical Society
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• v.24 no.2
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• pp.345-349
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• 2011

In this work, we find the genera of arithmetic subgroups $\langle{\Gamma}_{\Delta}(N),{\Phi}\rangle$ of $GL_2^+(\mathbb{R})$ generated by congruence subgroup ${\Gamma}_{\Delta}(N)$ and the Fricke involution $\Phi$.

• Journal of Veterinary Clinics
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• v.18 no.2
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• pp.139-145
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• 2001

This study was undertaken to determine the normal serial ultrasonographic appearance of the postpartum uterine involution in small pet dogs (Yorkshire terrier and Maltese). Postpartum changes in uterine shape, architecture, echogenicity and diameter were monitored by ultrasonography in 12 small pet dogs. Serial ultrasonographic examinations were done daily during the first week, 3 days interval from 8 to 30 days, and weekly from 31 to 100 days postpartum. The excretory period of vaginal discharge in 12 normal bitches of uterine involution was finished completely within 3 weeks postpartum. The short axis shape of the uterus was initially often flaccid-appearing. It varied from circular to polygonal. This lasted until 15.75$\pm$3.84 days postpartum, during which time the short axis uterine shape gradually changed to circular. Also, the long axis shape of the uterus was a beaded appearance until 30.89$\pm$4.25 days postpartum. After 30 days, it was appeared as tubular shape between placental and interplacental sites. The ultrasonographic image of the postpartum uterus consisted of four echogenicity distinct layers. Uterine wall was represented as very hyperechoic serosa, hypoechoic myometrium, hyperechoic endometrium and anechoic structures of fluid in the uterine cavity until 7 days postpartum. The individual uterine layers were most prominent during the first week postpartum, and they became progressively less distinct throughout the course of uterine involution. The thickness of myometrium was decreased rapidly in the placental sites from 4.47$\pm$1.42 mm at 1 day to 1.92$\pm$0.26 mm at 16 day, and in the interplacental sites from 3.19$\pm$0.61 mm at 1 day to 1.39$\pm$0.61 mm at 16 day. And it was decreased slowly until 94 day and was been minimum thickness at 94 day. The thickness of endometrium was also decreased like that of myometrium. The uterine diameter in the placental sites was decreased from 22.28$\pm$3.01 mm at 1 day to 16.11$\pm$1.46 mm at 7 day, and in the interplacental sites was decreased from 13.65$\pm$2.34 mm at 1 day to 9.41$\pm$1.59 mm at 7 day postpartum. From 7 day to 93 day, the change of diameter was more and more slow. At 94 days postpartum, the uterine diameter was 5~6 mm both placental and interplacental sites, and the uterine horns were uniform hypoechoic, tubular structures without enlargement. Therefore, complete involution of the uterus occurred at 94 days. It was concluded that normal post partum uterine involution in small pet dogs appeared to be completed 94 days postpartum by gross findings such as vaginal discharges, and by ultrasonographic findings, uterine shape and echogenicity.

• Asian-Australasian Journal of Animal Sciences
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• v.19 no.8
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• pp.1100-1105
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• 2006

During involution of the mammary gland after the lactation period, the gland undergoes an extensive epithelial cell death. In our previous study, overexpression of an extracellular proteinase inhibitor (Expi) gene accelerated apoptosis of mammary epithelial cells. Here we found that expression of the cathepsin D gene was induced in the Expi-overexpressed apoptotic cells. To understand the role of cathepsin D in apoptosis, we transfected cathepsin D gene into mammary epithelial HC11 cells and established the stable cell lines overexpressing the cathepsin D gene. We found that overexpression of the cathepsin D gene partially induced apoptosis of mammary epithelial cells. Expression patterns of the cathepsin D gene were examined in mouse mammary gland at various reproductive stages. Expression of the cathepsin D gene was increased during involution stages compared to lactation stages, and highest expression levels were shown at involution on day 4. We also examined expression of the cathepsin D gene in various mouse tissues. Mammary gland at involution on day 2 showed highest levels of cathepsin D mRNA of the mouse tissues that we examined. Liver tissues showed high levels of cathepsin D expression. These results demonstrate that cathepsin D may contribute to the apoptotic process of mammary epithelial cells.

• Animal cells and systems
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• v.4 no.4
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• pp.347-352
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• 2000

Programmed cell death, apoptosis, is a mechanism to maintain tissue homeostasis. Although the apoptotic process in rodent mammary tissues has been known to occur at the onset of involution, little is known about programmed cell death in the bovine tissues. Therefore, the purpose of this study was to investigate the molecular and cellular basis of apoptotic process in bovine mammary cells. Mammary tissues were obtained at different lactational and involurional stages. By apoptosis in situ endlabeling assay, apoptotic cells were found around the acinar celt lining in regressing bovine mammary tissues. The apoptosis-related genes bel-2 and bax were detected throughout involution by Northern blotting assay. The level of bax mRNA was dominantly expressed during involution. On the other hand, the bel-2 RNA transcripts were constantly expressed by 14 of post-lactation and declined thereafter. The expression of the testosterone-repressed prostate message-2 (TRPM-2) RNA transcripts, a marker for tissue remodeling, was increased as involution progressed. TNF a, were induced the DNA fragmentation and enhanced the expression of bax mRNA. In addition, milk protein secretion and amino acid uptake were decreased in mammary acinar culture treated with TNF $\alpha$. These results indicate that bovine mammary cells undergo apoptotic process after the cessation of milking and that TNF $\alpha$ may trigger apoptosis in lactating bovine mammary acini.

• Bulletin of the Korean Mathematical Society
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• v.53 no.4
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• pp.1157-1169
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• 2016

We find the distributional solutions of the Wilson's functional equations $$u{\circ}T+u{\circ}T^{\sigma}-2u{\otimes}v=0,\\u{\circ}T+u{\circ}T^{\sigma}-2v{\otimes}u=0,$$ where $u,v{\in}{\mathcal{D}}^{\prime}({\mathbb{R}}^n)$, the space of Schwartz distributions, T(x, y) = x + y, $T^{\sigma}(x,y)=x+{\sigma}y$, $x,y{\in}{\mathbb{R}}^n$, ${\sigma}$ an involution, and ${\circ}$, ${\otimes}$ are pullback and tensor product of distributions, respectively. As a consequence, we solve the $Erd{\ddot{o}}s$' problem for the Wilson's functional equations in the class of locally integrable functions. We also consider the Ulam-Hyers stability of the classical Wilson's functional equations $$f(x+y)+f(x+{\sigma}y)=2f(x)g(y),\\f(x+y)+f(x+{\sigma}y)=2g(x)f(y)$$ in the class of Lebesgue measurable functions.

• Journal of the Korean Mathematical Society
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• v.36 no.5
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• pp.959-1008
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• 1999

This paper considers foundational issues related to connections in the tangent bundle of a manifold. The approach makes use of second order tangent vectors, i.e., vectors tangent to the tangent bundle. The resulting second order tangent bundle has certain properties, above and beyond those of a typical tangent bundle. In particular, it has a natural secondary vector bundle structure and a canonical involution that interchanges the two structures. The involution provides a nice way to understand the torsion of a connection. The latter parts of the paper deal with the Levi-Civita connection of a Riemannian manifold. The idea is to get at the connection by first finding its.spary. This is a second order vector field that encodes the second order differential equation for geodesics. The paper also develops some machinery involving lifts of vector fields form a manifold to its tangent bundle and uses a variational approach to produce the Riemannian spray.

• Journal of the Korean Data and Information Science Society
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• v.23 no.1
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• pp.71-77
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• 2012

This paper introduces the projection methods by nonlinear regression model. To predict the student numbers, a log model and an involution model as the kind of a trend-extrapolation method are used. Empirical evidence shows that a projection by log model is better than by involution model with the confidence interval estimations for the coefficients of determination.

• Kyungpook Mathematical Journal
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• v.62 no.1
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• pp.43-55
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• 2022

Let R be a *-ring and n ≥ 1 be an integer. The objective of this paper is to introduce the notion of n-skew centralizing maps on *-rings, and investigate the impact of these maps. In particular, we describe the structure of prime rings with involution '*' such that _*[x, d(x)]_n ∈ Z(R) for all x ∈ R (for n = 1, 2), where d : R → R is a nonzero derivation of R. Among other related results, we also provide two examples to prove that the assumed restrictions on our main results are not superfluous.

• Communications of the Korean Mathematical Society
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• v.38 no.3
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• pp.679-693
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• 2023

Let 𝓡 be a 𝜎-prime ring with involution 𝜎. The main objective of this paper is to describe the structure of the 𝜎-prime ring 𝓡 with involution 𝜎 satisfying certain differential identities involving three derivations 𝜓₁, 𝜓₂ and 𝜓₃ such that 𝜓₁[t₁, 𝜎(t₁)] + [𝜓₂(t₁), 𝜓₂(𝜎(t₁))] + [𝜓₃(t₁), 𝜎(t₁)] ∈ 𝒥_Z for all t₁ ∈ 𝓡. Further, some other related results have also been discussed.